Where Did We Stop Last Time?

Cross-Sectional Data

Panel Data

Crime Rates and Probability of Arrest

• The above data can be loaded with

  data(crime4,package = "wooldridge")

• They are from C. Cornwell and W. Trumball (1994), “Estimating the Economic Model of Crime with Panel Data”.

• One question here: how big is the deterrent effect of law enforcement? If you know you are more likely to get arrested, will you be less likely to commit a crime?

• This is tricky: Does high crime cause stronger police response, which acts as a deterrent, or is crime low because deterrent is strong to start with?

• This is sometimes called a simultaneous equation model situation: police response impacts crime, but crime impacts police response

$$\begin{array}{cc}police& ={\alpha }_0+{\alpha }_{1}crime\\ crime& ={\beta }_0+{\beta }_{1}police\end{array}$$

Crime Rates and Probability of Arrest

Crime vs Arrest in Cross Section

css = crime4 %>% 
  filter(county %in% 
           c(1,3,145, 23))  
ggplot(css,
       aes(x = prbarr, 
           y = crmrte)) + 
  geom_point() + 
  geom_smooth(method="lm",
              se=FALSE) + 
  theme_bw() +
  labs(x = 'Probability of Arrest', 
       y = 'Crime Rate')

Crime vs Arrest in Cross Section: Positive Relationship!

xsection = lm(crmrte ~ prbarr, css)
coef(xsection)[2]  # gets slope coef

##     prbarr 
## 0.06480104

Ok, but what does that mean?

• Literally: counties with a higher probability of being arrested also have a higher crime rate.

• So, does it mean that as there is more crime in certain areas, the police become more efficient at arresting criminals, and so the probability of getting arrested on any committed crime goes up?

• What does police efficiency depend on?

• Does the poverty level in a county matter for this?

• The local laws?

• 🤯 wow, there seem to be too many things left out of this simple picture.

Crime in a DAG

Crime in a DAG

Within and Between Variation

You will often hear the terms within and between variation in panel data contexts.

Within and Between Variation: Give us a Plot.

Pooled OLS recovers between variation

• Let's add the mean of prbarr and crmrte for each of those counties to the scatter plot!

• And then a regression through those 4 points!

Accounting for Grouped Data: Introducing the Fixed Effect

OVB, IV and Panel Data

We've seen unobserved variable bias (OVB). For example, if the true model read:

$$y_i={\beta }_0+{\beta }_1{x}_i+c_i+u_i$$ if $c_i$ unobservable and $Cov(x_i,c_i)\ne 0\Rightarrow E[u_i+c_i|x_i]\ne 0,\text{with}{u}_i+c_i$ total unobserved component.

Dummy Variable Regression

mod = list()
mod$dummy <- lm(crmrte ~ prbarr + factor(county), css)  # i is the unit ID
broom::tidy(mod$dummy)

## # A tibble: 5 × 5
##   term              estimate std.error statistic  p.value
##   <chr>                <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)        0.0449    0.00456      9.87 9.85e-10
## 2 prbarr            -0.0284    0.0136      -2.08 4.86e- 2
## 3 factor(county)3   -0.0250    0.00254     -9.82 1.07e- 9
## 4 factor(county)23  -0.00850   0.00166     -5.13 3.41e- 5
## 5 factor(county)145 -0.00650   0.00160     -4.07 4.70e- 4

Dummy Variable Regression

First Differencing Solution

If we only had $T=2$ periods, we could just difference both periods, basically leaving us with

$$\begin{array}{cc}{y}_{i1}& ={\beta }_1{x}_{i1}+c_i+u_{i1}\\ y_{i2}& ={\beta }_1{x}_{i2}+c_i+u_{i2}\\ & \Rightarrow \\ y_{i1}-y_{i2}& ={\beta }_1(x_{i1}-x_{i2})+c_i-c_i+u_{i1}-u_{i2}\\ \Delta y_i& ={\beta }_1\Delta x_i+\Delta u_i\end{array}$$

where $\Delta$ means difference over time of and to recover the parameter of interest ${\beta }_1$ we would run

lm(deltay ~ deltax, diff_data)

The Within Transformation

The Within Transformation in R: Manual Solution

This works for our problem with fixed effect $c_i$ because $c_i$ is not time varying by assumption! hence it drops out:

$$y_{it}-{\overline{y}}_i={\beta }_1(x_{it}-{\overline{x}}_i)+c_i-c_i+u_{it}-{\overline{u}}_i$$

It's easy to do yourself! First let's compute the demeaned values:

cdata <- css %>%
  group_by(county) %>%
  mutate(mean_crime = mean(crmrte),
         mean_prob = mean(prbarr)) %>%
  mutate(demeaned_crime = crmrte - mean_crime,
         demeaned_prob = prbarr - mean_prob)

Then, run both models with simple OLS:

mod$xsect <- lm(crmrte ~ prbarr, data = cdata)
mod$demeaned <- lm(demeaned_crime ~ demeaned_prob, data = cdata)

The Within Transformation in R: Manual Solution

Interpreting the Within Estimates

• How to interpret those negative slopes?

• We look at a single unit $i$ and ask:

if the arrest probability in $i$ increases by 10 percentage points (i.e. from 0.2 to 0.3) from year $t$ to $t+1$, we expect crimes per person to fall from 0.039 to 0.036, or by -7.69 percent (in the reference county number 1).

Fixed Effects Estimation in R: use a Package!

• In real life you will hardly ever perform the within-transformation by yourself

• and use a package instead!

• There are several options (fixest is fastest). In our context:

mod$FE = fixest::feols(crmrte ~ prbarr | county, cdata)

• Notice the similar setup to the estimatr::iv_robust two-part formula. Here the fixed effects come after the |.

• Also, we can have more than one fixed effect! For a cool example with three fixed effects see the package vignette

Fixed Effects Estimation in R: use fixest 🙂

Within Transformation Animated